Assembling the Model Solution

I use College Board released AP items often in my Statistics course. The problems are aligned to clearly-stated goals, and the solutions provide insight not only into the grading of AP questions but also allow students to study well-articulated explanations. You can visit the College Board Statistics website and explore. Jason Molesky’s website provides helpful guidance on using FRAPPY’s (Free-Response AP Practice…Yay!) as a formative assessment tool in AP Stats.

Each free-response solution begins with the “model solution” – the ideal explanation a student would provide for full question credit. It is not unusual for Statistics students to struggle with clear communication, and having students read and dissect the model solution can be helpful in strengthening statistical arguments. A few times this year, I have used the model solution as a formative assessment tool with an activity I call “Assembling the Model Solution”.

Here’s how it works – start with an AP Free-Response question with a narrative aspect. Today, I chose a problem which requires students to interpret a P-value, from 2009:


The model solution contains a number of non-negotiable elements: a conditional probability, a reference to smple results, and the “extremeness” of results.


Next, I took the model solution as broke it into small, strategic “bites”. At the same time, I added some parallel distractors and a junk phrase or two.


Then, use a paper cutter and slice the Word document into phrase slices, and paper clip together. All students then received the problem and the slips of paper, with the challenge to assemble the model solution for part a of the problem.


The conversation were rich, and the teams mostly debated the salient aspects of the problem apprpriately. The biggest points of debate and incorrect solutions came from:

  • The difference between “sample” and “population” proportions.
  • The assumption of sameness in the treatments as the conditional aspect of P-value.

I have used this strategy a few times now, and continue to tweak how I provide the slips of paper. I’m also looking at digital options, but I like the social aspect of moving the slips of paper. The method is not ideal for everything in AP Stats, but there are a few areas in our curriculum where this fits in nicely:

  • Sampling and experimental design
  • Conclusions for inference procedures
  • Describing distribuitions.

You can download my file for this activity here.  Enjoy!

  • Credit to Jon Osters and the AP Stats glitterati who rightfully pointed out that my original post spelled “Yay!” incorrectly.

Today, Nothing Worked Well…and That’s OK

My 9th grade class has a quiz on statistics concepts tomorrow – standard deviation, interpreting graphs, outliers and the normal distribution. It’s a real cornucopia of stats ideas! To review, today’s class goal was to collect class-wide data using a fun applet, share using the collaboration space in OneNote, use a website to assess the data, and write our statistical summaries. A fun day filled with stats fairies and pixie dust! Here was the lineup:

  • shapesCollect data using Shapesplosion – an online game (think the old Perfection Game) developed by folks from Grinnell College. The plan was to play with, and without color. Aside: it’s OK if you disappear for a while to play with this site, it’s super-fun!
  • Share data using the collaboration space on OneNote.
  • Use the web apps to make graphs and produce statistical summaries.

This is what I had in mind….Here’s what really happened

  • Shapesplosion didn’t work – while I rehearsed the site on my laptop, it didn’t work for the kids. It was a Flash issue, and stopping to figure this out wasn’t in the cards. After a few minutes of hemming and hawing, I settled upon a far less fun data collection idea: Tell me a temperature you deem “cold” when you go outside, and one you deem “hot”. Not nearly as sexy as the time data I wanted…but hey, I needed a data set.  But at least we have data until…..
  • ArtOfStat was glitchy and wasn’t playing nice with copy/paste from OneNote. Kids are getting restless, we haven’t done much stats review, and I am definitely starting to lose my “big” class.

manuel.gifSo, what do you do when a lesson goes south, your objective is slowly slipping away and the kids smell chum in the water?


It’s not the kids’ fault when your plans go kaput. You may feel like some yelling is in order, but breathe, calm down, and be honest about what went wrong.

Student learning can’t be compromised because things go south. “There’s no time” is an easy out when we get rushed, but maintaining lesson fidelity is far more important than rushing to get to “stuff”.

Maintain clear expectations. Eventually all of my students were able to review some, and I had to alter my plan of attack. But stopping class, making sure we were all on the same page and understood the statistical expectations was necessary.

It won’t be the last time stuff goes wrong….roll with it…and laugh along with it.

Cocoa Puffs and Shared Work

Shared worked problems! What a magical time to be alive! What wonders does the magic algebra worksheet have for us to enjoy today?


OK….so most shared work problems suck. I apologize to my students aspiring to be pipe organ re-varnishers, but we can do so much better.

This week I used Cocoa Puffs, stopwatches and Desmos to bring some engagement to my rational expressions lessons. To start, each student was provided with a plate filled with 30 grams of Cocoa Puffs (incuding the plate) . After my 3-2-1 countdown, students picked Puffs one at a time from the plate and tossed them onto an empty plate.  As they completed the task, times were recorded for each student.

After students finished, I had them partner up and consider the question: “if you worked together with your partner on this task, with one plate of Cocoa Puffs, how long would it take you?”

Students asked a number of clariying questions (yes, there is one plate. yes, you can pick them off the plate together.), partnerships developed a few ideas. We debated the validity of many of them:

  • Many groups took the average of the two times, then divided the result by 2. This seemed reasonable to a number of groups, and led to a discussion of the vavlidity of averaging rates.
  • Some groups attempted to find a rate per gram. This was a good start, but given that groups did not know the mass of the plate (I use Chinet, so it’s bulky!), this introduced some guesswork.

To steer discussion, we focused on one student who took 80 seconds to complete the task. How much of the job did they complete after 40 seconds? After 20?  Can we write a function which depends on time here?  What does it mean? Crossing the bridge from the task time (80 seconds) to the job rate (1/80 per second) is a tricky transit. Using Desmos to show the “job” function lends some clarity.

desmos graph

From here, many partnerships felt more comfortable with establishing their own estimates.  The next day, teams shared their work and estimates on OneNote, then peer-assessed the communication.  Some of the work was wonderful, well-communicated, and served as a model for the class to emulate.

student work

puffsThe next day, we listed our calculated shared work predictions on the board, and tested our estimates. Teams timed each other with cell phone stopwatches, and did not let participants see the clock until the task was complete.



Many groups were quite close to their calculated predictions! We discussed why our predictions didn’t quite meet the actual – bumping, variability in mass, general panic – and when error is acceptable. And now we have a firm background in rates and rational functions – time to conquer those pipe organs!